Let P(X,A) be the number of positive divisors of A not greater than X; for example, P(1,12)=1, P(2,12)=2, and P(12,12)=6.
1)If P(12,A)=7 and P(A,A)=10, what's A?
2)If x*y is a divisor of A, then show that
P(x*y,A) ≥ P(x,A)+P(y,A)1
Since we know A has 10 positive divisors, we know it is of the form p^4*q^1 for primes p and q. Since 7 of these divisors are 12 or less it makes sense that p and q be relatively small. Trying 2^4*3^1=48 works perfectly.
P(12,48)=7 {1,2,4,8,3,6,12}
P(48,48)=10 {1,2,4,8,16,3,6,12,24,48}
This P is an interesting function. I think a harder version of part 1 would be nice to see.
I haven't started part 2 although its obviously true if x and y are primes and x*y=A. 4=3+21
Edited on July 17, 2007, 9:40 am
Edited on July 17, 2007, 9:41 am

Posted by Jer
on 20070716 10:57:36 